The LCoS SLMs can only display a quantized phase and spatial profile, approximating an ideal blazed grating due to the finite pixel size (spatial quantization) and due to the limited available phase values (phase quantization). Higher orders (m ≠ 1) are generated and unintentionally coupled to other output positions, which is denoted as static crosstalk. Not only the quantization of phase and pixel, but also other physical effects such as fringing field effect, phase flicker, and device non-uniformity, would induce an error in the phase profile. Such deviation from an ideal blazed grating would thus generate higher orders, i.e., static crosstalk.

Another kind of crosstalk occurring only during switching is called transient crosstalk [

43,

44]. During switching, the phase pattern is not controlled intentionally; thus for a short time period, the transiently generated diffraction orders will be coupled to other output ports. Mitigation approaches have been proposed, such as inserting an intermediate phase pattern with an un-periodic pattern between the start and end gratings of the switching process or using the complex addressing sequence during the switching [

45]. These approaches address the fundamental cause of the transient crosstalk by disturbing the periodic phase structure so that the light is randomly scattered instead of diffracted during switching

In order to fully understand where the static crosstalk comes from and how it could be reduced, the LCoS device, GAEA from Holoeye [

46] (10 megapixels resolution, pixel size 3.74 µm, digitally addressed backplane), is studied, calibrated for wavelength 1550 nm (with default voltage setting for the low and high electrode levels to be 0.5 and 1.5 volts). Below, two physical phenomena, phase flicker and fringing field effect, are illustrated and their relations with device level crosstalk are discussed.

#### Phase Flicker

A digital pulse width modulation scheme is used in current displays as the driving sequence for representing different gray levels [

47,

48]. Due to the finite viscosity of the LC molecules, the time-averaged voltage is observable for LC molecules, which is related to phase representation; however, the superimposed pulse modulation pattern produces certain fluctuation in the orientation of the LC molecules, which leads to the flicker [

49] on the beam of light. This effect could be detrimental for images as the gray level is drifting around the desired value; thus such uncertainty could reduce the diffraction efficiency of gratings [

50]. By using a higher frequency for the driving sequence, the flicker amplitude can be reasonably reduced, which is also demonstrated by Martínez et al. for visible bandwidth [

51].

Figure 2 shows the experimental setup for flicker measurement, in which linearly polarized light vibrating at 45° with respect to the LC director (the LC director is parallel to the long axis of the display for GAEA device) impinges perpendicularly onto the entrance window of the device. Different gray levels are addressed to the display under the default voltage setting (the low and high voltages on the electrode are 0.5 and 1.5 volts, respectively, enabled by the control software of the GAEA device). The state of polarization (SOP) of the reflected light is measured with a Stokes polarimeter, which provides the time-averaged stokes parameters for a time interval longer than the characteristic flicker period. In

Figure 3 and

Figure 4, we show the experimental results obtained for the GAEA by applying the averaged Stokes polarimetric technique demonstrated in [

52,

53].

This averaged Stokes polarimetric technique is described as follows; the GAEA device is a parallel-aligned LCoS device (PA-LCoS), thus it can be considered equivalent to a variable linear retarder, the retardance of which varies as a function of the applied voltage (gray level). This algorithm for the retardance calculation is based on Mueller-Stokes formalism and models the linear variable retarder including retardance instabilities (flicker), wherein the fluctuation of retardance (flicker) is approximated as a triangular time-dependent profile. By measuring the Stokes parameters (SOPs) for the input and output light and for all grey levels displayed on the device, the retardance and flicker parameters can be obtained by fitting the theoretical expressions and experimental values for each gray level.

Figure 3 is the measurement result for the Stokes parameters of the output reflected light. S1, S2, and S3 are the measured stokes parameters of the output light versus the gray level. DoP is short for ‘degree of polarization’, which denotes how much light is polarized.

Average retandance as a function of the gray level (i.e., the calibrated look-up table (LUT) of the GAEA device) is calculated by the above mentioned method and is plotted in

Figure 4. The LUT provides the information with which the gray level must be addressed to the screen so that the desired phase value is written onto the incident light wavefront. From

Figure 4, we see that the presented LUT of the GAEA device is quite linear, which indicates that the device is being well calibrated under the default voltage setting. The flicker value is different for each grey level and the maximum flicker is about 35° for gray level 120.

One way of reducing the flicker is to lower the temperature of the LCoS device. It has been proved that a reduction of up to 80% of the flicker is possible when the LCoS is brought to −8 °C [

54]. Therefore, temperature control electronics can be added to the backplane of the SLM to decrease as well as stabilize the operating temperature so the impact of temperature drift is minimized.

#### Fringing Field Effect

When the pixel spacing is smaller than the thickness of the LC layer, the electric field is no longer homogeneous over a pixel. Due to the inhomogeneous distribution of the electrical field across a single pixel, the phase response is also not constant upon a pixel. Several researchers have illustrated this fringing field effect [

55], which is considered the main limitation to LCoS performance. A width/height ratio,

τ_{E}, is used as an indicator of how strong the fringing field is. It is defined in Equation (2), where W is the width of the electrode and d is the LC cell thickness:

In one simplified model of the fringing field effect by Uzi Efron [

56], where the amplitude modulation is almost ignored in the modeling, the blazed grating diffraction efficiency is given quantitatively by:

where

η is the blazed grating diffraction efficiency, Λ is the length of the grating period, and ∆

X_{FB} is the fly-back zone width, which is defined as the broadening width of the phase profile, particularly in areas of sharp spatial transition such as blaze resets. This equation is widely used for fast calculation of the expected diffraction efficiency. In another model built by Lu et al. [

57], the near field phase profile of the grating is observed under a microscope. The profile is then fitted using the error function, and the diffraction efficiency is calculated by the angular spectrum method. This method provides a way of diffraction efficiency optimization based on the near field phase profile optimization instead of the much more typical far field optimization algorithm related to a computer generated hologram (CGH) [

58,

59,

60]. The near field optimization method provides a more direct and accurate measurement. Similar near field approaches [

61] are also proposed by measuring the sub-pixel Jones matrices and modeling the fringing field effect as a low pass filter.

In order to compensate for the diffraction efficiency reduction of the blazed grating due to the fringing field effect, a voltage profile optimization method has been verified using rigorous numerical simulation software for liquid crystal devices by X. Wang et al. [

62,

63]. Especially for small pixel devices, the deformation of the phase profile is huge compared with that of ideal blazed grating in the phase reset region. As we can see from the simulation result by X. Wang et al, the efficient modulation depth for a blazed grating is not able to achieve 2π when using the LUT obtained for the uniform screen due to the fringing field effect. Thus, optimization of the diffraction efficiency could only be done by changing the voltage applied to the electrodes. By changing the voltage of each pixel in a blazed grating iteratively, the phase value on each electrode could be adjusted to be similar to that of the desired phase profile. He also analyzed the relationship between the fringing field effect and various parameters given by rigorous simulation such as pixel size, cell thickness, the electrode spacing, the voltage profile, the gap between electrodes, the birefringence of the LC material, the pretilt angle, the elastic constants, and the surface alignment direction. As shown by the simulation result, a high birefringence material is critical for wide-angle LC optical phase array for better performance regarding the diffraction efficiency. Experimental verification of the voltage optimization method has also been performed. In one study by E. Haellstig [

64], a LC SLM with 1 × 4096 small stripe shaped pixels (1.8 µm) was studied. By using the voltage profile optimization method, the diffraction efficiency could be improved significantly.

We would like to see whether this voltage optimization method for the diffraction efficiency of blazed gratings would be useful for the GAEA we have. Further experimental verification is done and presented in the following. In the case of diffraction efficiency measurements, we use a slightly modified version of the experimental setup in

Figure 2, where a lens is added at the output of the LCoS to focalize the diffracted orders on the lens focal plane. Light incidents perpendicularly to the GAEA device. The input light is linearly polarized parallel to the LC director. We display blazed gratings with different numbers of pixels per period, each pixel corresponding to a phase level. The first order diffracted intensity is measured with a power meter. This data is presented as the original blazed grating in

Figure 5. We then apply the voltage optimization method. After voltage optimization, the first order diffracted light of the blazed gratings is measured again, and we can see that the diffraction efficiency increases dramatically, especially for the small period gratings (large diffraction angel). The measurement data is plotted in

Figure 5 as a function of the diffraction angle, calculated with the grating equation [

65], where Λ is the grating period,

λ is the wavelength in use, and

θ is the diffracted angle.

As we can see from

Figure 5, the above-mentioned voltage optimization method is proven to be effective for integer periods of blazed gratings, especially for LCoS with small period sizes (large diffraction angle). Without considering other factors such as the output fiber position, the pure crosstalk generated by the LCoS device is represented by the diffraction efficiency of other higher orders. Thus, higher diffraction efficiency for first order diffracted light also indicates a lower diffraction efficiency for other orders, i.e., lower crosstalk for the system. As we can see, the diffraction efficiency for the optimized grating increases with respect to the non-optimized as the diffraction angle increases; this indicates that the crosstalk is reduced compared with original result.

For non-integer periods of blazed grating, due to the inherent large crosstalk induced by the grating structure itself (even by the ideal phase profile), although the voltage optimization method is proved to be effective, the crosstalk between different channels is still relatively high.

For next generation high resolution displays, it is desired not to have ‘crosstalk’ between pixels, i.e., no fringing field effect. Thus, methods have come up by, for example, inserting a polymer wall between pixels (15 µm pixel size is demonstrated [

66]) or having three electrodes in one pixel to generate a homogenous electric field [

67].